Department of Mathematics
Course Outline for Mathematics 181/281
Dynamical Systems
Spring 2019
Time MWF 9:00 AM - 9:50 AM
Place: Millikan Room 2393
Instructor:
Dr. Adolfo J. Rumbos
Office: Andrew 2287.
Phone/e-mail: ext. 18713 /
Courses
Website: http://pages.pomona.edu/~ajr04747/
Office
Hours: TR 9:00 am - 10:00 am, or by appointment
Texts: (Recommended) Ordinary
Differential Equations by Jack K. Hale
Dover Publications, Inc., 2009
edition.
(Recommended)
Nonlinear
Dynamics and Chaos by Steven
H. Strogatz
CRC Press LLC, 2nd edition.
Prerequisites: Linear algebra, elementary ordinary differential
equations and some real analysis course
Course Description. This
course provides an introduction to the theory and applications of continuous
Dynamical Systems. We begin with the
study of the fundamental existence and uniqueness theorems for ordinary
differential equations, as well as the results on continuous dependence on
initial conditions and parameters. We
then define continuous dynamical systems and explore their properties. In particular, we study stability and
long-term properties of dynamical systems, and discuss various techniques and
results that arise in the theory and applications.
Course
Structure and Expectations. The structure of the coursed is centered on
lectures and readings on the topics listed in the attached tentative schedule
of lecture and examinations, homework assignments, two examinations and a term
project. Readings and problem sets will be assigned
and collected on a weekly basis.
Students are strongly encouraged to work on every assigned problem. Late homework assignments will not be
graded. The term project will
consist of a paper and presentation
on a topic not covered in the lectures.
Ideas for topics in the term project may be found in the recommended
texts for the courses; possible topics may range from applications of the
theory and techniques learned in class to problems in various fields in science
to advanced analysis techniques that are not covered in the course. Presentations will take place in the last
weeks of the semester
Grading
Policy. Grades will be based on the homework, two examinations
and a term project involving an advanced topic in the theory and applications
of dynamical systems. The overall score
will be computed as follows:
homework 20%
Examinations 50%
term project 30%
Math 181/Math
281 Spring
2019
Tentative Schedule of Lectures and
Examinations
Date Topic
W Jan.
23 What is a dynamical
system
F Jan.
25 Fundamental existence
and uniqueness theory
M Jan. 28 Existence
and uniqueness theory continued
W Jan.
30 Existence and
uniqueness theory continued
F Feb.
1 Continuous dependence
on initial conditions
M Feb. 4 Continuous
dependence on initial conditions (continued)
W Feb.
6 Continuous dependence on parameters
F Feb.
8 Continuous dependence on parameters (continued)
M Feb. 11 Global
existence results
W Feb.
13 Global existence
(continued)
F Feb.
15 Global existence
(continued)
M Feb. 18 Definition
of dynamical systems
W Feb.
20 Integral curves, flow
domains and flows
F Feb.
22 Integral curves, flow
domains and flows (continued)
M Feb. 25 Orbits
W Feb.
27 Invariant sets
F Mar.
1 Singular points
M Mar. 4 Review
W Mar.
6 Exam
1
F Mar.
8 Fixed points
M Mar. 11 Invariant
sets
W Mar.
13 Cycles and periodic
solutions
F Mar.
15 Cycles and periodic
solutions (continued)
M Mar. 18 Spring Recess!
W Mar.
20 Spring Recess!
F Mar.
22 Spring Recess!
Date Topic
M Mar. 25 Equilibrium
points
W Mar.
27 Liapunov
stability
F Mar.
29 Cesar Chavez Day
M Apr. 1 Liapunov
stability (continued)
W Apr.
3 Liapunov functions
F Apr.
5 Liapunov Theorem
M Apr. 8 Limit
cycles
W Apr.
10 Stability of limit
cycles
F Apr.
12 Planar systems
M Apr. 15 Planar
systems (continued)
W Apr.
17 The Poincaré-Bendixson
Theorem
F Apr.
19 The Poincaré-Bendixson
Theorem (continued)
M Apr. 22 Review
W Apr.
24 Exam 2
F Apr.
26 Special Topic
M Apr. 29 Special
Topic
W May
1 Special Topic
F May
3 Special
Topic
M May 6 Special Topic
W May
8 Special Topic